12
Part of 2005 AMC 10
Problems(2)
Trefoil
Source:
1/14/2009
The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length ?
[asy]unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(12pt));pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180);
pair E=B+C;draw(D--E--B--O--C--B--A,linetype("4 4"));
draw(Arc(O,1,0,60),linewidth(1.2pt));
draw(Arc(O,1,120,180),linewidth(1.2pt));
draw(Arc(C,1,0,60),linewidth(1.2pt));
draw(Arc(B,1,120,180),linewidth(1.2pt));
draw(A--D,linewidth(1.2pt));
draw(O--dir(40),EndArrow(HookHead,4));
draw(O--dir(140),EndArrow(HookHead,4));
draw(C--C+dir(40),EndArrow(HookHead,4));
draw(B--B+dir(140),EndArrow(HookHead,4));label("2",O,S);
draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar);
draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar);[/asy] (A)\ \frac13\pi\plus{}\frac{\sqrt3}{2} \qquad
(B)\ \frac23\pi \qquad
(C)\ \frac23\pi\plus{}\frac{\sqrt3}{4} \qquad
(D)\ \frac23\pi\plus{}\frac{\sqrt3}{3} \qquad
(E)\ \frac23\pi\plus{}\frac{\sqrt3}{2}
geometry
Probability of Product Being Prime
Source:
1/14/2009
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
probability